I understand the point about empathy, but all qualified professionals will be aware of this and should find ways to connect and build relatedness, irrespective of race. I really don't know if there will be much advantage although I do accept it could work that way. And it could also be the opposite where a same-race psychiatrist assumes a natural empathy and makes no further effort on it, whereas a different-race psychiatrist notes the cultural gap and focuses hard on rapport building.
Personally I would consider the beliefs/methodology of the psychiatrist as a more important factor. In particular, how readily they will opt for drug/electrotherapy treatment instead of persevering with talking therapies.
IMHO the talking/psychological therapies are under-used or seen as too time-consuming/difficult, and therefore people are put onto medication when it might not be needed. Don't get me wrong, drug and/or electrotherapy and definitely appropriate in many cases, but when I look at the usage (and growth in usage) of anti-depressants, I just find it staggering.
To the OP, I would suggest asking the psychiatrist their views on different treatment methods too before deciding. Also see the below link if it's of use, wish you all the best !
http://www.hpb.gov.sg/HOPPortal/health-article/10210